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Doofus writes "The Atlantic has an interesting story about opening up what we routinely consider 'advanced' areas of mathematics to younger learners. The goals here are to use complex but easy tasks as introductions to more advanced topics in math, rather than the standard, sequential process of counting, arithmetic, sets, geometry, then eventually algebra and finally calculus. Quoting: 'Examples of activities that fall into the "simple but hard" quadrant: Building a trench with a spoon (a military punishment that involves many small, repetitive tasks, akin to doing 100 two-digit addition problems on a typical worksheet, as Droujkova points out), or memorizing multiplication tables as individual facts rather than patterns. Far better, she says, to start by creating rich and social mathematical experiences that are complex (allowing them to be taken in many different directions) yet easy (making them conducive to immediate play). Activities that fall into this quadrant: building a house with LEGO blocks, doing origami or snowflake cut-outs, or using a pretend "function box" that transforms objects (and can also be used in combination with a second machine to compose functions, or backwards to invert a function, and so on).' I plan to get my children learning the 'advanced' topics as soon as possible. How about you?"

I'd point out: I can't do single-digit arithmetic without errors. I never have been able to. I can do math in my head pretty decently; one time on a road trip I got bored, and came up with a km/miles conversion ratio starting from a vague recollection of the number of inches in a meter. I came up with 1.61. Google says 1.60934.

But give me a hundred single-digit addition problems, and I will get a couple of them wrong.

Yes, it's called embryonic development. It affects millions of people around the world and leads to impaired math abilities, where the affected cannot handle hundreds of mental calculations before making an error. The only known cure is to spend years in a basement alone eating cheetos, while insulting others' trivial math and lingual mistakes.

I like you ideas, and would like both to subscribe to your newsletter, and build a train wreck of a web site where a multitude of these embryonically impaired people can co-mingle and share fantasies about Natalie Portman.

No to both, and that's actually a sort of stupid question, given the obvious evidence of general competence at larger-scale arithmetic, which I achieve in part by doing calculations twice using different paths and confirming the results.... and still do it significantly faster than most people. So far as I can tell, it's just a tuning thing; my neurons are tuned for faster responses rather than more-reliable responses, so I get answers quickly but sometimes they're wrong.

Doing the same thing 100x is only "simple but hard" if you can actually do it accurately.

I agree. But I disagree with TFA's comment about "simple but hard".

Repetitive != Hard

Once you understand the concepts then doing 100 problems is no more difficult than doing 10. It just takes 10x longer to finish them all.

And that is the purpose of assigning a large number of tasks. Someone who does NOT understand the concept can work through 10 problems in an hour. Someone who DOES understand the concepts can do 10 problems in a minute.

So the 100 problem task is used to find those who did not finish because they did not have time because they did not understand the concepts.

Any teacher handing that out to someone who can already do it isn't doing their job properly.

Yes. Once they've completed the 100 problem task the first time they've shown that they've mastered the concepts so they can move on.

But we've become so focused on getting a grade (A, B, C...) for doing the work that we've lost sight of WHY we were doing the work in the first place.

And that is the purpose of assigning a large number of tasks. Someone who does NOT understand the concept can work through 10 problems in an hour. Someone who DOES understand the concepts can do 10 problems in a minute.

When I started being home-schooled (for health reasons, not religious reasons), my Mom bought Saxon math books. They may still have a large number of problems, e.g. 100, but they mix up old types of math problems with newly learned types. That way I didn't forget old learning and I was less bored, while still learning new material.

When I started being home-schooled (for health reasons, not religious reasons), my Mom bought Saxon math books. They may still have a large number of problems, e.g. 100, but they mix up old types of math problems with newly learned types. That way I didn't forget old learning and I was less bored, while still learning new material.

So you are saying that you would have trouble completing 100 addition problems right now because it would be "hard" for you?

Yes. I would have trouble completing the 100 assigned problems, as I'd throw the paper in the trash and go do something more interesting. It is "hard" to sit for 10 minutes in a boring mindless repetitive task.

So the 100 problem task is used to find those who did not finish because they did not have time because they did not understand the concepts.

You're assuming that the speed at which the problems are solved is positively correlated with fundamental understanding of the concepts. For problems like multiplication, this isn't really the case. Someone who memorizes the "times tables" may have a less complete understanding of the concept but finish quicker.

This is the flaw in timed math assignments with a large number of problems. It penalizes taking time to think about the problems and come up with the correct answer in favor of rote memorization.

Absolutely, memorization can be a very useful tool and tasks that exercise memory skills should be part of school curricula. It's just that it shouldn't be taught instead of the fundamental concepts of a subject, which is what I think happens when timed multiplication tests are given too soon.

And I'm glad I didn't waste my time with multiplication tables. Math is not about speed, and making it about speed and memorization just gives people a fundamental misunderstanding of what it's about, and chases away some of the elite few who would otherwise be capable of truly understanding it. I happen to know that 9 x 9 = 81, but it's not because I made an explicit effort to memorize that; it happened naturally, simply because I saw the result many times.

You're assuming that the speed at which the problems are solved is positively correlated with fundamental understanding of the concepts. For problems like multiplication, this isn't really the case.

Not only is it not the case, in highly intelligent people, for large problem sets, it is often reverse-correlated. When I was a kid, if you gave me 50 math problems, I'd take longer to solve them than the folks who were making Fs in the class—not because I was struggling, but because after the first five o

I would think the context was sufficient to clarify which of the several meaning of hard was being used.

By your own example doing 100 problems requires 10x as much time and mental energy doing 10. What word would you use to describe the increase in labor? Clearly digging a swimming pool with a spoon is qualitatively different than digging a seed-hole.

I had a horrible time with math in grade school, especially multiplication - my brain just doesn't store trivia well: 7*6 =....? Couldn't tell you offhand.

It is harder to stand in a 3 hour line than it is to stand in a 3 minute line. If they come out and state that the line is expected to take 3 days rather than 3 hours, many will leave, unable to do it. Focus on quantity of rote work at the expense of concepts is wrong, especially in this day and age of Mathematica, Maxima, etc...

Yes and no. There are certainly some benefits to repeated practice in developing the speed and accuracy of computations. The problem is that some people may never master these low-level computations due to undiagnosed cognitive disabilities (i.e. discalculia or problems with working memory) and this content is being used as a gate-keeper to higher-level mathematics which the person could potentially master with appropriate support. Different types of mathematical activities use different areas of the bra

The problem is that some people may never master these low-level computations due to undiagnosed cognitive disabilities (i.e. discalculia or problems with working memory) and this content is being used as a gate-keeper to higher-level mathematics which the person could potentially master with appropriate support.

Yes. And that is why the tasks should be used to identify those who have not mastered the concepts instead of just to assign a grade.

Not so sure. I knew a guy who had an MA (from Oxford, no less) in Maths and he absolutely sucked at mental arithmetic; he could never have worked as a bartender in the days before the tills did the magic for you.

On the other hand he knew many conceptual things that I'd never even heard of before.

That's not to say that learning your tables is useless; it's precaching commonly used calculations and burning them into your ROM.

I am sorry, but if you think that repetitive arithmetic helps with intuitive sense for math, then I must admit I think you are stupid, or you fail to comprehend "intuitive" sense. I've done a lot of tutoring of Maths and Physics over the decades. Math majors have an inferior intuitive sense of probability theory than do business majors. The ability to parrot a proof, or calculate for an hour without making a sign error, has nothing to do with understanding. Sometimes understanding what something is, and

There's an assumption that repetition will help recollection. I don't think it's entirely wrong, though of course you can overdo it.

The reason why you need recollection is so you can see the patterns.

Suppose someone tells you "multiply any integer by 5, and the last digit is always a 5 or a 0". How are you going to get a sense of whether that's true if you don't have at least few results to hand? Now, this isn't rigorous proof, but it is mathematical intuition. Any number of mathematical observations will s

I disagree. Digging a trench with a spoon is "simple but hard", because each spoonful is simple, but knowing that a shovel would work better makes all the extra hours painful. The question to ask is do students need to learn rote skills before being introduced to concepts, or can they be taught concepts as soon as they are able to understand them? Seems obvious to me...

I had that argument given to me at school. In practice it is rubbish.Just buy a bloody calculator.

I would say knowing stuff like long division is valuable, however I do not know it off the top of my head and have never needed to do it since school.If I had to do it now I'd figure out how it worked from what I remember and *then* do the problem.

When I was a kid Mrs. Dunn (one of the parents of a kid at the school) taught an optional "math club" a half hour before school on Wednesdays. I don't remember exactly what we learned (it's since merged with all the "real" math classes I took), but I do remember learning sumnation and some other fairly advanced concepts.

Kids are smart, and they are totally capable of learning a lot of advanced math.

The trick is getting to kids before their idiot peers who casually go around saying things like "Math is hard", "I can't do math, it's difficult", "Math is only for really super smart people."

Math is actually pretty easy, but once you've convinced yourself it's hard it becomes twice the battle, first to get past that mental barrier about how impossible it is.

Same applies to many areas of study. I was coding like a coding fool on National Coding Day and my High School counselor wouldn't let me into the programming classes because my math grades needed to be higher. Pfft, like math is more prevailing than logic. Anyway, plenty of misconceptions on what people are really capable of, particularly at a very young age.

I think there's a growing culture of morons who think you should molly coddle kids rather than get those little brains working during the time in their lives when they are capable of learning the fastest.

Calculus, taught properly, is incredibly easy and intuitive because it's all geometry - you can teach it visually, with no numbers.

Area under a curve? No harder to understand qualitatively than the area of any other shape. Slope of a curve at a point? Again, quite easy to understand with construction paper cut-outs of curves, and a ruler.

And there are plenty of real physics problems that can be solved with simple geometry! Make a drawing of velocity over time that tells a story of a trip. With constant acceleration, all the shapes will be triangles and rectangles. Find the area to find the distance travelled.

For actual curves, you can make them from wood and weigh them to find the integral. Awesome hands-on fun that completely de-mystifies calculus. Not sure a kid would be ready for it by 5, but 8-10, no problem.

Doing hands on geometrical calculus is easy, and can be understood quite easily. What's I actually found difficult, was not the concept, but the memorization of how go obtain the integral or derivative of a functions. So many rules, that seemingly had no logic to them. The derivative of sin(x) is cos(x). Why? most students probably couldn't tell you that. Looking at a proof I found [math.com], it actually seems quite non-obvious, and not something most beginner calculus students could figure out on their own.

Ahh. I have the answer to that one! The answer is the same as "why does e^(pi * i) = -1", in a very non-obvious way, but it's very simple.

Why is the derivative of e^x = e^x? Because that's what makes e special - we picked 'e' to make that true. if you look at exponential curves for various bases, it becomes clear that somewhere between 2 and 3 this neat thing happens, and it turns out to be quite handy. If you play around with a graphing calculator it becomes obvious that it must be true for some number, and you can observe/discover "oh, that's e - so that's why its called the natural log".

Why is the derivative of sin(x) equal to cos(x)? Because we use radians. If you measure angles in degrees or grads or whatever, it doesn't work out this way. But if you study simple harmonic motion (which back in the days if record players everyone did), or just think about a point moving around a circle as viewed edge-on, it you will observe/discover that there's this neat property something moving that way: it's velocity as seen edge on is the same as it's position as seen edge on, rotated 90 degrees.. This is really visually obvious with a toothpick stuck to the outside of the spinning platter of a record player!

Once you grok that visually, then clearly there must be some way of measuring angles such that the derivative of sin(x) is cos(x), because that's what those functions mean: the position as viewed from the side, and the position as viewed from the side after rotating 90 degrees! It just so happens that choosing the range [0 2pi) for angles makes the math work out properly. Proving why it's 2pi and not some other value, like proving why it's e and not some other value, is a mess, but you can just observe that some such value must exist for both cases.

While I probably didn't pick the best example, as looking at the graphs of sin(x) and cos(x) you can see that one is the derivative of the other, there are plenty of more complicated rules out there. Polynomials are quite easy as well. But once you get into more complicated functions applying all the rules can be frustrating. Often the questions are more about reducing a function to something else that's easily derivable than about how to actually find the derivative.

Sure, but by that point you're doing computation, not learning the principle involved. Few people find doing computation to be the fun or interesting part of math, which is why we automate it. Doing enough exercises to be good at it, like memorizing multiplication tables, is worthwhile eventually, but it's a terrible place to start.

I had problems with things like integration by parts. I couldn't see how it worked, and something inside wouldn't let me just learn it by rote. Being off injured during that part of the course didn't help, also.

I always found differentiation easier than integration. Is that objectively true, or is it just me?

Why is the derivative of sin(x) equal to cos(x)? Because we use radians. If you measure angles in degrees or grads or whatever, it doesn't work out this way.

I'm sorry, what? If you plot the two functions and look at the slope (derivative) of one compared to the value of the other, the relationship will be the same whether you label the x axis as "degrees", "radians", "grads", or "blutarskis", as long as the conversion is a simple multiplicative factor (as is degrees to radians, etc.)

I.e., d/dx sin(nx) = cos(nx) because you can replace nx with y by assigning y = nx. Then you have d/dx sin(y) = cos(y) which we know is true.

Calculus, taught properly, is incredibly easy and intuitive because it's all geometry - you can teach it visually, with no numbers.

The problem isn't that the students are dumb, but that the teachers aren't allowed to teach to the children in the class. The materal and methods are set for the state, and there's little negotiation available.

My calculus teacher was useless - I learned calc from my physics teacher, who was free to teach calculus any way she pleased. (But then, she always said she'd quit if she was ever forced to teach a specific way by the state, and eventually she did quit because of that.) But's that's school, which is sort of off-topic in a thread about learning.

I plan to make sure my children understand what they're taught, and are taught new things based on what they already know. If that means teaching them complex ideas earlier than they would normally learn them then that's fine, but to make that a goal in itself is nonsensical.

I have always wondered why puzzles were never included in any educational system. Logical puzzles, spatial manipulation, patterns, and lateral thinking challenges go a long way towards improving general intelligence and learning abilities. Much more so than, say, memorizing multiplication tables. It also helps them with those complex ideas that you spoke of.

Instead, kids are taught to hate math and hate puzzles, and standardized tests are a joke.

I have always wondered why puzzles were never included in any educational system. Logical puzzles, spatial manipulation, patterns, and lateral thinking challenges...

My kids attend public school in California. Their math assignments regularly include puzzles of all the types you mention, as well as other recreational math, often adopted directly from the grand master [wikipedia.org].

Instead, kids are taught to hate math and hate puzzles, and standardized tests are a joke.

My kids like math, enjoy the puzzles, and the standardized tests (at least in math) are quite good (and often include questions requiring insight, that most people would consider "puzzles").

For me, having been introduced to the basic idea of a "hard" concept made it a lot easier when the subject was taught in school ten years later. For example, basic cooking introduced me to a lot of math and a little chemistry. At age five, making lemonade was age-appropriate. It made sense that to make half as much lemonade, we'd use half as many lemons. (Ratios). Gee, we used one cup of sugar to make a big jug of lemonade, how much sugar should we use to make half as much? In school, fractions were e

Exactly. One of the best things my parents did for me while I was growing up was provide "out-of-band" education of that variety. They'd introduce a concept without any of the trappings that typically surround a math lesson, giving me nudges and having me intuit how the concept worked, without putting any pressure on me to learn it right then. If I did, great, but if I didn't, no worries. It made the in-class lessons that came later on significantly easier, since they were just a formalized restatement of concepts that I already understood.

Aside from basic arithmetic, the stuff I pull out of my math toolbox the most often would have to be the way that geometry and calculus taught me to view the world. There aren't many opportunities to FOIL binomials in everyday life*, but if I have some scrap wood and need to figure out how to get the most out of it for a project, geometry has taught me a load of different ways to dissect that shape. If I have a problem that needs to be broken down for an algorithm, the basic idea behind integration (that you can take infinitely small cross sections and sum them together) has numerous applications. If I need a rough approximation of a volume, that same concept can be applied in my head in a few seconds, without any need for busting out a pen and paper or for remembering all of the dx/dy specifics.

And, really, much of that can be taught to kids at a young age. They don't need the "math" of it, so much as they need that way of viewing the world, and you can teach people at a young age how to break down things in those sorts of ways so that they can have an intuition for how things add up, without having to explain sigma notation or whatnot. When they learn integration by parts later on, they should have an "well of course it works that way" attitude, rather than the "wait, you can do that?!" attitude most people learning it seem to have.

* Funny story. I was at a Thanksgiving get-together a few months back, and a high schooler I know came by to ask me for help with her algebra II homework, since her parents hadn't been able to help and I was one of the people there with the most math lessons under my belt. I was able to help her to a point, but a lot of that stuff was just beyond my recollection since the last time I had used it was 15 years prior when I learned it, and without a textbook or other reference guide there, I wasn't able to help. In swoop about a dozen college students to the rescue...or so I thought. In talking it over with them, however, all of them either got stuck at or before the place that I got stuck, so I found myself working with them to try and reformulate the problem using calculus. Finally, a college freshman saved the day, since she had taken algebra II just a year or two prior and still remembered the thing we were all missing. Point is, it was pretty obvious that none of us had used that part of algebra II in the years since we were taught it, whereas calculus was something we all felt much more comfortable applying, despite the fact that it's supposed to be harder.

I remember being in grade school and being irritated that for the 3rd year in the row I was learning how to do basic math. Then when I got to high school I was pissed off that I was rushed though from algebra to trig in 4 years. I don't think they understood that basic math is easy and higher math is hard and your math level has nothing to do with your grade level.

The reason that you were irritated is because you were one of the smart kids. I felt the same way in school, until one day a teacher told me that they weren't constantly reviewing the basics for *me*. They were doing it for the other 90% of the kids in the class who weren't like me.

If my parents had been able to afford a private school, or if I had access to a "gifted" school, it would have been different (and much better). But in a public school, you can't fault teachers for having to teach to the lowest c

Back when I was in grade school, probably grade 4 or 5, there was this reading comprehension system. It had a bunch of colored levels, and on each level there would be ten booklets. Each booklet had a story and a question sheet. You would mark your answers on an answer sheet using the same color pencil crayon as the level you were on.

They had something similar for mathematics [nationalst...tre.org.uk] back when I was in primary school, except that rather than 10 booklets there were dozens of cards. The teacher would assign each pupil 10 cards, and then we could do them in the order we wanted (as long as no-one else was using the card we wanted). I loved it.

My school had a one afternoon per week gifted students program. Among other things we did programmed/self paced instruction and classroom work on boolean algebra and basic number theory. This was in the late 1960s in a middle class school district in suburban Pittsburgh (Avonworth.)

The other thing worth noting is how most mathematicians make their breakthrough discoveries before age 30. (Sorry don't have the reference for this, but I've seen it widely discussed.) So that means the earlier we expose kids "with the math gene" to more complex topics, the greater the possibility that stuff will 'stick'.

in first grade there are pre-algebra and problem solving concepts being taught now. at least in my kid's public schoollast night i had a huge argument with him about the proper strategy to use to solve a problem. i had to google the common core lesson plans to help him

In my experience, with young children your best chance at teaching them these things is to relate it to their current interest. My 4 year old is really into maps right now, he draws me one every day at his preschool. I've been showing him different maps and trying to relate the concept of directions etc. With his interest in drawing hopefully I can work in the alphabet at some point too. It's a tricky task to put things in terms a 4 year old mind and attention span can digest without overwhelming them.

This article does not contain any description of calculus-like activities that five-year-olds are participating in. There's a lot of 'this is cool' commentary without any description of what 'this' actually is.

That's so true. You start out with all kinds of high goals for you kids and by the time they are teenagers you are just happy if they stay out of trouble and will be able to take care of themselves when they are an adult.

I've said it before, but kids already do simple algebra in elementary school.

3 + [] = 5

and you fill in the box.

Yes and no. In one sense, that's an algebra problem, but not all elementary students are taught to solve it *as* an algebra problem.

I've often seen that problem given to a child like this: "three plus what is five? Come on, three, plus something, is five. What's the something? I have three, and if I add this many more, I get five..." That's not algebra, that's guessing. The child is often thinking "is it one? No. Is it two? Three plus two is five. Yes, its two."

I think that one of the problems with the way math is taught in schools is the fact that very little is done to explain how calculations students are doing can be applied to actual problems. Now that I'm older, went through a science education in college and work in a technical field, I understand this. However, one of my problems early on was that I never really felt comfortable doing math problems. It sounds really stupid, but I must have some sort of disability -- I can't do basic arithmetic in my head.

If you can get a student to understand what you mean when you say exponential growth, and how it relates to something they care about, then students will understand it more.

At my school that would have been the one where you're walking along a road, and at the first pub you have one sip of beer, at the second two, at the third four etc; how far down the road do you get before you fall over.

Dr. Maria Montessori, who before becoming a doctor and then an educator, was an engineering major and loved the math portion of it. Thus in her method that she devised 100 years ago, five-year-olds learn the 3D-geometric equivalent [cabdevmontessori.com] of binomials and trinomials from high school algebra.

In the interests of balance, if you work in an open plan office that's her fault.

On the contrary, the open floor plan has its roots [scientificamerican.com] in the 20th century philosophy of Modernism combined with a focus on industrial efficiency by early 20th century industrialists. Maria Montessori, in contrast, adapted traditional values to the modern era. The multiple ages grouped together doing work simulates the traditional large family (plus cousins). One of the problems she was addressing was the dual-working-parents lea

If you want to prepare children for higher level mathematics and all that learning it implies, please start with logic. The idea of teaching young kids calculus is a bit absurd and not nearly as helpful as a foundation in logic. When you have a malleable mind that is still growing and rapidly changing giving an early foundation in how to think critically and how to approach abstract questions would seem to have a larger benefit than having them think about calculus.

Even if they don't get it the first time, continued exposure is good. I can think of a lot of things in math that didn't "click" until I'd heard it the umpteenth time. For example, how to count to umpteen.

I think a little bit of "modern" math is good but the old stuff still needs to be taught. Rote memorization gets a bad rap; but IMHO the 10X10 multiplication table should be committed to memory just like the alphabet. All else equal, a student with the table in his head will be able to work more quickl

Rote memorization gets a bad rap; but IMHO the 10X10 multiplication table should be committed to memory just like the alphabet. All else equal, a student with the table in his head will be able to work more quickly and confidently than one without.

I agree. There are some who don't. Occasionally you see people posting here who think it's a from of brainwashing & that learning a trick like 9 x 8 is 10 times 8 less 1 x 8 somehow makes them Thomas Paine. Of course to do that you still need to know what 1

Teaching Calculus to five year olds is stupid. But that's not really what the article describes. I think a critical distinction implied by the article but not stated is that there's a difference between rigorously teaching a topic to the point of mastery, and exposing children to a topic to make them familiar and comfortable with it.

I've always believed that 50% of time in school should be spent rigorous teaching, and 50% of the time should be spent easing students into more complex topics over time. I t

What a shameless and ridiculous headline. 5 year olds can't even usually read... or count above 100. I just got my 6 year old to understand that 0 comes before 1 for gods sakes and he's the smartest kid in his class. If building legos is calculus than I'm a god damned genius. WTF is this even about?

The article didn't make this terribly clear, but people seem to be missing the point.

If you teach the concepts through hands-on interactive play, kids as young as five can understand the concepts underlying Calculus without too much difficulty. This also happens to be one of the best times in your life for learning, when the brain is rapidly forming new connections.

Her point is teach the concepts, teach the patterns, teach kids how to find patterns, and how to internalize mathematical knowledge.

The mechanical drudgery of formal language, writing out and solving equations, etc comes later on but builds on the fundamental understanding developed much earlier in life.

that's what teachers call "timed tests." Very popular because easy to prepare, conduct, and grade. But getting into stuff like the number line, proportions, ratios, rates of change, etc. it becomes abstract. However, I wish I was given the number line and also do graphs in elementary school instead of waiting for college. I mean a number line that shows negative numbers. No need to get into complex graphs but can do stuff like plot quantities of stuff compared to other things.

As the homeschooling parent of a 5 year old we have learned this first hand. We stumbled upon a set of books called Life of Fred that are "story books" that incorporate math. They were written by a math professor tired getting students that didn't know math and thought it was "hard". He incorporates basic algebra using x from almost the very beginning. They cover many topics that most think of as "advanced math" in simple, natural ways. As the story unfolds Fred has to use math in a variety of situations. It shows that math is practical and teaches it in an accessible way.
Even better, the stories are silly and ridiculous and fun for all ages.

I recalled an/. article from 4 years ago with a completely different view of maths for children.Here it is [slashdot.org]
Basically, during the depression Boston needed to make cuts to the public schools, so they cut maths from all of the schools in the poor neighborhoods until 6th grade. By 7th grade all of the students who only had 1 year of maths were at the level of the students who had 6 years.

It makes some sense to me, math is really just logic, and a child's brain is not wired for logic. Though, part of me also thinks that "math is a young man's game" and you need a way to identify the geniuses before it's too late.

Is it just me, or is the education system getting far too concerned with "keeping children engaged" and "making learning fun", than actually teaching concepts.

You don't only teach memorization of addition/multiplication tables in order for the child to know their multiplication tables. You do it because that sort of rote memorization (especially of abstract items) is good for the brain. Children also need to learn that a lot of work is actual work, and some of it involves fairly boring mental drudgery.

Exactly, let them play. What they are saying is to let them play with things like mirror books. They will see patters there and their brain will learn things while playing. Then when the time comes, years later, to actually learn the math, their brain has some sort of reference to relate to that makes it easier.